Cycles UV/Image Editor: How can I control the Image Displayed?
UV/Image Editor Window 


Cycles Render has texture nodes. Different than Blender Render.  This candidate answer may be biased towards Blender Render and may be deleted.
You may also switch to Blender Render from Cycles temporarily for a time period. Requires mouse movement and Click.  Perhaps only slightly more effort and humiliation than going into edit mode.  Some small amount of testing.  Oh, yes, ummm and do not forget to switch back to Cycles, otherwise the results could be ... Catastrophic .  Another Click.




Above image shows a pinned and unpinned image.  One is static. One follows the current mesh object in Blender Render Edit Mode, Not Cycles.  Thus we can see as many images as we want.



Viewport Shading. 2 simultaneous 3D Views.  Top is Solid Shading.  Bottom is Material Shading.  Image above.  Same Mesh.  Local Mode is meant conserve CPU/GPU resources in this contrived example.

Techniques to see the current texture for an object


Blender Render. When you enter edit mode for the Mesh, Blender makes an effort to change the image.  Please investigate. You can quickly menu into and out of edit mode.  Just confirming the OP original statement .... Cycles Render Failed this test. Even when I selected the texture node in the Node Editor Window.  This selection action has some effect in other parts of Blender such as Baking.
If computer resources permit, then you can have a 2 or more ... 3D Views, one with Material displayed. Local Mode may control CPU resource consumption.  More control with display options.  Or Switch Between Material and Non Material in Single 3D View. 
Blender allows multiple windows with Menu/Window/Duplicate Window. You need not flood your current window as forbidden in a comment below.  This would again allow multiple 3D Views in the own window. Window meaning here is different than a 3D View.  
You can open up 2 or more ... Image Windows. Pin the result of one by enabling "Display current image regardless of object selection". Enter mesh edit  mode with one object, choose the image window non pinned and select a new image.  You can see two images at once.  At least until you enter edit mode on a different mesh.
A single quick frame render may be a reminder.


Blender Behaviour allows flexibility because of So Many Potential Uses


The Image Window has many uses. Blender can not always make the correct guess .  The Blender User must have control.  Blender automatically changing the image for the Blender User May be contrary to the finicky and excessively moody whims of the user.
The little menu highlighted in your question allows you to change the image displayed or show the render result.
Perhaps you just want to inspect for fun, one of your many images.
Image Window can display the new result of Texture Painting so the result image can have no partner mesh. Odd Situation. Please search Texture Painting if that suits you.
Blender Render. You can have many images associated with a Mesh as  Material Textures in the Texture Panel.  The Blender User must be able to inspect the exact image that User desires at that moment.  Even if the images are displayed in the Material Window or Texture Panel.